Vector Fields

The standard way to represent \(N\)-dimensional vector fields in tntorch is via a list of \(N\) tensors, each of which has \(N\) dimensions. Functions that accept or return vector fields do so in that form. For example, gradient():

import torch
import tntorch as tn

t = tn.rand([64]*3, ranks_tt=10)
grad = tn.gradient(t)
print(grad)

[3D TT tensor:

 64  64  64
  |   |   |
 (0) (1) (2)
 / \ / \ / \
1   10  10  1
, 3D TT tensor:

 64  64  64
  |   |   |
 (0) (1) (2)
 / \ / \ / \
1   10  10  1
, 3D TT tensor:

 64  64  64
  |   |   |
 (0) (1) (2)
 / \ / \ / \
1   10  10  1
]

We can check that the curl of any gradient is 0:

curl = tn.curl(tn.gradient(t))  # List of 3 3D tensors
print(tn.norm(curl[0]))
print(tn.norm(curl[1]))
print(tn.norm(curl[2]))

tensor(1.0344e-06)
tensor(1.4569e-06)
tensor(2.5995e-06)

Let’s also check that the divergence of any curl is zero (we’ll use a random, non-gradient vector field here):

vf = [tn.rand([64]*3, ranks_tt=1) for n in range(3)]
tn.norm(tn.divergence(tn.curl(vf)))

tensor(4.8832e-08)

… and that the Laplacian of a scalar field t equals the divergence of t’s gradient:

tn.norm(tn.laplacian(t) - tn.divergence(tn.gradient(t)))

tensor(0.)